While the implications of a \(Z_{\cal N}\)  shift symmetry to solve the strong CP problem (with a \(1/{\cal N}\)  probability and \({\cal N}\)  degenerate worlds) have been previously analyzed [1]}, leading to a lighter-than-usual axion, the question of DM and the cosmological evolution was left unexplored.
We showed here that the evolution of the axion field through the cosmological history
departs drastically from both the standard one and from previously considered mirror world scenarios.
